{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:32Z","timestamp":1753893812314,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A non-empty set ${\\cal F}$ of $n$-bit vectors over alphabet $\\{0,1\\}$ is called singly repairable, if every vector $u \\in {\\cal F}$ satisfies the following conditions:(i) if any bit of $u$ is changed (from $0$ to $1$ or vice versa), the new vector does not belong to ${\\cal F}$(ii) there is a unique choice of a different bit that can then be changed to give another vector $\\neq u$ in ${\\cal F}$.Such families ${\\cal F}$ exist only for even $n$ and we show that $2^{n\/2} \\leq |{\\cal F}| \\leq {2^{n+1} \\over (n+2)}$.  The lower bound is tight for all even $n$ and we show that the families of this size are unique under a natural notion of isomorphism (namely, translations and permutation of coordinates).  We also construct families that achieve the upper bound when $n$ is of the form $2^m-2$.  For general even $n$, we construct families of size at least $2^n\/n$.  Of particular interest are minimal singly-repairable families. We show that such families have size at most $2^n\/n$ and we construct families achieving this upper bound when $n$ is a power of $2$. For general even $n$, we construct minimal families of size $\\Omega(2^{n}\/n^2)$. The study of these families was inspired by a computational scheduling problem.<\/jats:p>","DOI":"10.37236\/1956","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T05:08:53Z","timestamp":1578719333000},"source":"Crossref","is-referenced-by-count":0,"title":["Combinatorics of Singly-Repairable Families"],"prefix":"10.37236","volume":"12","author":[{"given":"Eugene M.","family":"Luks","sequence":"first","affiliation":[]},{"given":"Amitabha","family":"Roy","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2005,11,15]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r59\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v12i1r59\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:40:04Z","timestamp":1579322404000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v12i1r59"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2005,11,15]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2005,1,7]]}},"URL":"https:\/\/doi.org\/10.37236\/1956","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2005,11,15]]},"article-number":"R59"}}